9Model parameters (Symposium, 2005)Finite-population assumed to be generated from a superpopulation modelInference on model parameterTotal variance of :: model expectation and variance: design expectation and variancei) if f ≈ 0 thenii) if f ≈ 1 thenwhere f is the sampling fraction. For multistage sampling, the psu sampling fraction plays the role of f.In case i),

10Example: Ratio estimator when y is assumed to be randomforDefineWe havewhere Ad is a 2×N matrix of random variables with kth column:We getwhere Ab is a 2×N matrix of arbitrary real numbers with kth column:where is an estimator of the total variance of

11Estimator of the total variance ofandwhenA variance estimator of is given bywithwhereNote that is an estimator of model covariancewhen and when

17Simulation 2: Conditional performanceWe generate R=20,000 finite populations , each of size N=393 from the ratio modelusing the number of beds asOne simple random sample of size n=100 is drawn from each generated populationParameter of interest:We arranged the 20,000 samples in ascending order of values and then grouped them into 20 groups each of size 1,000

22Simulation 3: Estimating equationsWe generated R=10,000 finite populations , each of size N=393 from the modelUsing the number of beds asleads to an average of about 60% for zOne simple random sample of size n=30 is drawn from each generated populationParameter of interest:Population units are grouped into two classes with 271 units k having x<350 in class 1 and 122 units k with x>=350 in class 2Post-stratification: X=(271,122)T